3.136 \(\int \frac{1-\sqrt{3}+x}{x \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=138 \[ \frac{2}{3} \left (1-\sqrt{3}\right ) \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

(2*(1 - Sqrt[3])*ArcTan[Sqrt[-1 - x^3]])/3 + (2*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(
1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3
] + x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1
- x^3])

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Rubi [A]  time = 0.11038, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2}{3} \left (1-\sqrt{3}\right ) \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{2 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - Sqrt[3] + x)/(x*Sqrt[-1 - x^3]),x]

[Out]

(2*(1 - Sqrt[3])*ArcTan[Sqrt[-1 - x^3]])/3 + (2*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(
1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3
] + x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1
- x^3])

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Rubi in Sympy [A]  time = 12.9296, size = 121, normalized size = 0.88 \[ \left (- \frac{2 \sqrt{3}}{3} + \frac{2}{3}\right ) \operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x-3**(1/2))/x/(-x**3-1)**(1/2),x)

[Out]

(-2*sqrt(3)/3 + 2/3)*atan(sqrt(-x**3 - 1)) + 2*3**(3/4)*sqrt((x**2 - x + 1)/(x -
sqrt(3) + 1)**2)*sqrt(-sqrt(3) + 2)*(x + 1)*elliptic_f(asin((x + 1 + sqrt(3))/(
x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(3*sqrt((-x - 1)/(x - sqrt(3) + 1)**2)*sqrt(-
x**3 - 1))

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Mathematica [A]  time = 1.31409, size = 156, normalized size = 1.13 \[ \frac{2}{3} \left (-\sqrt{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\tan ^{-1}\left (\sqrt{-x^3-1}\right )-\frac{3 \left (\sqrt [3]{-1}-x\right ) \sqrt{\frac{x+1}{1+\sqrt [3]{-1}}} \sqrt{\frac{\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}} \sqrt{-x^3-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - Sqrt[3] + x)/(x*Sqrt[-1 - x^3]),x]

[Out]

(2*(ArcTan[Sqrt[-1 - x^3]] - Sqrt[3]*ArcTan[Sqrt[-1 - x^3]] - (3*((-1)^(1/3) - x
)*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*Sqrt[((-1)^(1/3) - (-1)^(2/3)*x)/(1 + (-1)^(1/3
))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(S
qrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]*Sqrt[-1 - x^3])))/3

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Maple [A]  time = 0.019, size = 135, normalized size = 1. \[ -{\frac{2\,\sqrt{3}}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{2}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }-{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x-3^(1/2))/x/(-x^3-1)^(1/2),x)

[Out]

-2/3*arctan((-x^3-1)^(1/2))*3^(1/2)+2/3*arctan((-x^3-1)^(1/2))-2/3*I*3^(1/2)*(I*
(x-1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((1+x)/(3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x-1/
2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3-1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x-1/2-1
/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(3/2+1/2*I*3^(1/2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - \sqrt{3} + 1}{\sqrt{-x^{3} - 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x),x, algorithm="maxima")

[Out]

integrate((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x - \sqrt{3} + 1}{\sqrt{-x^{3} - 1} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x),x, algorithm="fricas")

[Out]

integral((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x), x)

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Sympy [A]  time = 6.99394, size = 61, normalized size = 0.44 \[ - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} - \frac{2 \sqrt{3} i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{2 i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x-3**(1/2))/x/(-x**3-1)**(1/2),x)

[Out]

-I*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4/3)) -
2*sqrt(3)*I*asinh(x**(-3/2))/3 + 2*I*asinh(x**(-3/2))/3

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - \sqrt{3} + 1}{\sqrt{-x^{3} - 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x),x, algorithm="giac")

[Out]

integrate((x - sqrt(3) + 1)/(sqrt(-x^3 - 1)*x), x)